UNITS AND DIMENSIONS. M.Sathyanarayanan AP/CIvil-VCET

Transcription

1 UNITS AND DIMENSIONS

2 UNITS AND DIMENSION Measurement and Units Fundamental units Systems of units Dimensional Analysis

3 UNITS AND DIMENSION :: Why do we need units? We need units because we want to measure the Amount or quantity of some things. To make this measurement globally acceptable we need to put some Unique measurement value. This value is called a UNIT

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5 There are several units systems for measurement of physical quantities The most common systems are Metric, FPS, CGS and MKS system

6 Systems of Unit FPS: Foot, Pound, Second CGS: Centimeter, Gram, and Second MKS: Metre, Kilogram and second SI: System International

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8 Exa E peta P Tera T hecta h 10 2 Deka da 10 1 Deci d 10-1 Pica p 10-12

9 What are Units? Many people aren't sure of the difference. Let's try and get a set of definitions we can use. Consider 110 mg of sodium 24 hands high 5 gal of gasoline We'll break them up this way

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11 A "dimension" can be measured or derived. The "fundamental dimensions" (length, time, mass, temperature, amount) are distinct and are sufficient to define all the others. We also use many derived dimensions (velocity, volume, density, etc.) for convenience. "Units" can be counted or measured. Measured units are specific values of dimensions defined by law or custom. Many different units can be used for a single dimension, as inches, miles, centimeters, furlongs, meters and Kilometer are all units used to measure the dimension length.

12 Derived Units Physical Unit Unit Symbol Quantity Acceleration metre/second2 m/s2 Angular Velocity radian/second rad/s Angular acceleration radian/second2 rad/s2 Force N or Newton kgm/s2 Moment of Force Newton metre Nm Work, Energy Joule J or Nm Torque Newton metre Nm Power Watt W= J/s2 Pressure Pascal Pa = N/m2 Frequency Hertz Hz or 1/s

13 Units and Calculations It is always good practice to attach units to all numbers in an engineering calculation. Doing so attaches physical meaning to the numbers used, gives clues to methods for how the problem should be solved, and reduces the possibility of accidentally inverting part of the calculation. Addition and Subtraction Values MAY be added if UNITS are the same. Values CANNOT be added if DIMENSIONS are different.

14 EXAMPLES: different dimensions: length, temperature -- so cannot be added same dimension: length, different units -- can add Multiplication and Division Values may be combined; units combine in similar fashion. EXAMPLES: 4.5 is a "dimensionless" quantity (in this case a pure number) You cannot cancel or lump units unless they are identical.

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16 Dimensional Homogeneity Every valid equation must be "dimensionally homogeneous" (a.k.a. dimensionally consistent). All additive terms must have the same dimension. Dimensionless Quantities When we say a quantity is dimensionless, we mean one of two things. First, it may just be a number like we get when counting.

17 M! mass, L! length, T! time Dimensions of Some Common Mechanical Quantities M! mass, L! length, T! time Table 1.1 Dimensions of Some Common Mechanical Q Quantity Dimension MKS unit Angle dimensionless Dimensionless = radian Steradian dimensionless Dimensionless = radi an2 Area L2 m2 Volume L3 m3 Frequency T-1 s!1 = hertz = Hz Velocity L!T-1 m! s "1 Acceleration L!T-2 m! s "2 Angular Velocity T-1 rad! s "1 Angular Acceleration T-2 rad! s "2 Density M! L-3 kg! m "3 Momentum M! L!T-1 kg! m! s "1 Angular Momentum M! L2!T-1 kg!m2! s "1 Force M! L!T-2 kg m s -2 = newton = N Work, Energy M! L2!T-2 kg!m2! s "2 = joule = J Torque M! L2!T-2 kg!m2! s "2 Power M! L2!T-3 kg!m2! s "3 = watt = W Pressure M! L-1!T-2 kg! m "1! s "2 = pascal= Pa

18 Arithmetic and Dimensions There are strict rules for doing arithmetic with quantities that have dimension. 1. You can only add, subtract, or compare quantities with the same dimension. So, you can add two lengths, or add two masses, but you can t add a length and a mass. 2. You can multiply and divide quantities with any dimension. Anything goes with multiplication and dimension. 3. Sine, cosines, logarithms, etc. The input x in something like sin(x) or ln(x) or log(x) must always be dimensionless and unit less. No exceptions. 4. Exponentiation, for instance mb, involves two quantities: an exponent (b) and a base (m).

19 The exponent must be dimensionless. That is, it must be a pure number with no units. No exceptions. The base must also be dimensionless, unless the exponent happens to be an integer. Or, more precisely, the dimensions of the base have to match the exponent in a way that the result has sensible dimensions. So, (3miles) 2 is a perfectly sensible calculation, producing a result of 9 square miles. But 3miles is meaningless. That business above starting, Or, more precisely... means that it would be fine, for example, to take the square-root of 9 square miles. The result would be 3 miles. This will be clearer when you ve read about dimensions of M.Sathyanarayanan calculatedap/civil-vcet quantities.